design and optimization of tubular linear permanent-magnet

Scientia Iranica D (2020) 27(6), 3053{3065

Sharif University of TechnologyScientia Iranica

Transactions D: Computer Science & Engineering and Electrical Engineeringhttp://scientiairanica.sharif.edu

Design and optimization of tubular linearpermanent-magnet generator with performanceimprovement using response surface methodology andmulti-objective genetic algorithm

S. Arslana, O. G�urdalb, and S. Akkaya Oyc;�

a. Department of Electrical and Electronics Engineering, Harran University, Urfa, 63400, Turkey.b. Department of Electrical and Electronic Engineering, Faculty of Technology, Gazi University, Ankara, Turkey.c. Fatsa Faculty of Marine Sciences, Ordu University, 52400, Turkey.

Received 24 December 2017; received in revised form 13 July 2018; accepted 29 October 2018

KEYWORDSLinear generator;Permanent magnet;Finite element;Multi-ObjectiveGenetic Algorithm(MOGA);Optimization.

Abstract. Linear generators are electric machines which generate electrical energy fromlinear movement. Since these machines can lift gear wheel or power train, they arenowadays widely used. Considering the fact that the working areas of these machines di�erwith speed and power characteristics, this study deals with the design and optimizationof tubular linear generator for free piston practices. The design considered responsesurface optimization through variables that were acquired by sizing via the interface. Thecorrelation between the determined design variables and the magnitude of the generatoroutput was examined. In addition, the obtained amounts were used for objective functionsof increasing e�ciency, decreasing overall volume, and improving general performance andthe optimum values were found by using Multi-Objective Genetic Algorithm (MOGA).Initial and optimum design data were compared by ANSYS Maxwell 2D. With overallperformance improvement, 22.78% decrease in total mass and 11.7% decrease in cost wereobserved. In addition, a prototype for the linear generator was created in line with theinitial geometry data and applied by the crank slider mechanism.© 2020 Sharif University of Technology. All rights reserved.

1. Introduction

Considering the vast usage area of linear generators,e.g., free piston applications, wave energy, stirling sys-tem, shock absorbers, vibrators, compressors, mobilechargers, vibration energy harvesting devices, mobilelighting appliances, and space applications, the stud-

*. Corresponding author. Tel.: +9(0 414) 318 30 00;Fax: +9(0 414) 318 32 28E-mail addresses: [email protected] (S. Arslan);[email protected] (O. G�urdal);[email protected] (S. Akkaya Oy)

doi: 10.24200/sci.2018.50093.1506

ies on these generators have experienced increasingdevelopment day by day. Since linear machines aregenerated from rotating machines, their working prin-ciples are not di�erent. However, they are di�erentin terms of geometric structures, design equality, andmethods. As it is well known, electric machine design-ers should consider designing by taking demands suchas e�ciency, total weight or local weight, cost, powerdensity, etc. into account. In order to meet thesegoals, designers generally use optimization methods,e.g., pattern search, sequential nonlinear programming,Genetic Algorithm (GA), Response Surface Methodol-ogy (RSM), Monte Carlo simulation, etc., in packageprograms. ANSYS Maxwell is an instance that uses Fi-

3054 S. Arslan et al./Scientia Iranica, Transactions D: Computer Science & ... 27 (2020) 3053{3065

nite Element Method (FEM) the validity and reliabilityof which have been proven in the literature. Dalcali andAkbaba [1] examined the e�ect of parametric variationof pole arc o�set distance on the performance of apermanent-magnet synchronous generator. Bouloukzaet al. [2] performed optimization by using Monte Carlomethod. They showed that there was a good agreementbetween the ANSYS Maxwell 2D calculations and theanalytically calculated values of the optimum designof slotted Halbach Permanent-Magnet SynchronousMotor (PMSM). Qinghua et al. [3] performed theoptimization of PMSM by using RSM and developedthe prototype of the motor. They showed that thenumerical results obtained through this optimizationmethod and the application results matched. Ab-baszadeh et al. [4] performed the optimization of cavitygap and slide in order to decrease the cogging torquewith the help of RSM in brushless DC (BLDC) electricmotor. In the cogging torque of the optimized motor, asigni�cant decrease was seen in the simulation resultsby FEM. Jolly et al. [5] described their work on thedesign optimization of PMSMs using RSM and GAs.Ghasemi [6] used RSM to reduce the stroke forceof surface magnet synchronous motor. The resultsshowed that the optimum values of RSM were moree�cient than those of the GA and particle swarmoptimization in cogging torque reduction. Yu et al. [7]examined the e�ects of stator and rotor sizing variablesof embedded PMSM on cogging torque by RSM.They made initial and optimum motor comparisons.Bremner [8] examined the e�ects of the basic geomet-rical proportions of embedded PMSM on the machineperformance by RSM. Jabbar et al. [9] changed therotor dimensions of embedded synchronous motor andperformed optimization by using RSM and GA. Ab-baszadeh et al. [10] examined the e�ects of the basicgeometrical proportions of surface-insert permanent-magnet synchronous machine on machine performanceby RSM. Arehpanahi and Kashe� [11] used magnetostatic FEM analysis and RSM to reduce cogging torquein interior PMSM. Arslan et al. [12] performed theoptimization of a torus-type axial ux machine by usingRSM. They observed that the general performance ofthe machine was maximized and the cogging torqueand weight continued to decrease. Saha et al. [13]stated that e�ciency improvement could be e�ectivelyachieved by designing the optimization of line-startpermanent-magnet motor rotor structure using theRSM. Ahn et al. [14] proposed the approach as ane�cient way to improve the performance of optimallydesigned permanent magnetic actuator and to reducethe number of experiments. The optimum designof the dual-permanent-magnet-excited machine wasinvestigated by Jian et al. using RSM [15]. Hasanienet al. [16] performed optimization by using RSM andGA to improve the weight and thrust of transverse

ux linear motor. Pourmoosa and Mirsalim [17] pre-sented a low-speed single-sided linear induction motordesign. They performed optimization by using RSMto decrease motor weight and increase thrust. Theyfound that simulation and application data matchedto a great extent. In our previous studies on lineargenerator, Arslan et al. [18] constructed a linear gen-erator model through ANSYS Maxwell with analyticalequations designed in MATLAB GUIDE. They carriedout surface magnet tubular linear generator design andoptimization. In line with the objective functions theydetermined, they used pattern research algorithm andobtained optimum sizing. Arslan et al. [19] examinedmagnetic ux density of the change in stator and rotorparts of the linear generator. Arslan and Oy [20]performed inset magnet tubular linear generator designand optimization. They used GA and SequentialNonlinear Programming to reduce the cogging force.Wang et al. [21] performed optimization of tubularlinear motor by means of decomposition-based multi-objective di�erential evolution particle swarm. UsingMulti-Objective Genetic Algorithm (MOGA) alone canstretch to the area near an optimal Pareto front; how-ever, it requires more computing time than the multi-objective optimization approach does. Parallelizationcauses a considerable decrease in computing time ateach owchart stage [22]. In this study, the softwareANSYS Maxwell 2D was used in the calculations forthe design of the tubular linear generator. The e�ectsof sizing variables of the tubular linear generator onoutput variables were examined by means of correlationanalysis. In addition, ANSYS Workbench RSM andMOGA were used to obtain optimum sizing in linewith the objective functions determined. The initialdesign and the design obtained by optimization werecompared using FEM.

2. Analysis of the correlation between thevariable parameters of the generator ande�ciency, power out, weight, cost, andcogging torque

Regression is an approach to modeling the relationshipbetween two or more variables functionally. The valueof the variable y is estimated for the values of the inde-pendent variable x. Correlation is used to see whetherthere is relationship between two numerical variables,and if there is, to see the direction and size of thisrelationship. The mathematical model of the tubularlinear generator (Figure 1) is written in MATLAB GUI.Analytical sizing data are given in Table 1.

It is vital to identify the parameters that willhave direct e�ect on the performance of the generatorin the optimization process, since incorrectly chosendesign variable(s) has an undesirable in uence on thesuccess of optimization. In addition, the determined

S. Arslan et al./Scientia Iranica, Transactions D: Computer Science & ... 27 (2020) 3053{3065 3055

Figure 1. (a) 2D and (b) 3D representation of geometrical sizing of tubular linear generator.

Table 1. Generator input design parameters.

Parameter Value Unit Parameter Value Unit

Power 600 VA Slot pitch 0.01833 m

Frequency 50 Hz Lm 0.005 m

Stroke 0.0275 M G 0.002 m

Rated speed 2.75 m/s Tm 0.0198 m

Shear stress 2 N/cm2 Tw 0.009166 m

Current density 3 A/mm2 Hw 0.04473 m

Induced voltage 55 V De 0.160 m

Alfa (Tm/Top) 0.72 | Slot �ll factor 0.6 |

Beta (Bw/Tos) 0.5 | A coil winding number 76 |

Slot/Pole 6/4 | Approximate cost of copper 30 Tl

Winding factor 0.866 | Approximate cost of magnet 250 Tl

Lowest common multiplier 12 | Approximate cost of steel 10 Tl


input variables should not be associated with eachother, since this will lead to calculating the correlationbetween output parameters and input parameters. Itis important to �nd out the correlation between inputvariables and output parameters and to �nd out towhat extent an input variable explains the output pa-rameter. In line with this objective, optimization pro-cess will be more successful with suitably de�ned inputparameters. In addition, a brief look at the literaturein this area reveals that, generally, designers of electricmachines choose the parameters which have a �rst-degree in uence on the geometry of motor/generator.However, variables are generally considered to be innerdiameter, outer diameter, and pole pitch ratio. Inthis study, primary length and primary inner diameterwidths are regarded as stable. Primary variablesare considered to be air gap width (g), thicknessof magnet (Lm), slot-pitch ratio (Beta), pole-pitchratio (Alfa), and primary yoke ux density (Byp). Inorder to �nd out the correlation between these designvariables and design output parameters, Maxwell 2Drz linear generator model was developed by ANSYSWorkbench. Geometric variables and dimension sizeswere de�ned by design features. Control conditionswere identi�ed in Maxwell 2D rz. Input and outputparameters (stroke, e�ciency, power out, cost, volume,iron loss, etc.) were determined in the Maxwell 2Ddesign research section. Parameter correlation analysismodule was added. Research points were formed by thespeci�ed limits of input variables. Correlation analysismethod was determined. Design samples were solvedby the program until the speci�ed calculation criterionwas reached (Figure 4).

The correlation between input and output pa-rameters is expressed in matrix form with variableswhich can be positive or negative, ranging between +1and �1. The relationship is inverse with a negativevalue (i.e., as the input variable increases, outputdecreases with respect to the unit coe�cient). On theother hand, positive values indicate direct relationship(i.e., as the input variable increases, output increases

Table 2. Input variables and margins.

Inputvariable

Initialdesignvalue

Searchpoints

(Min-Max)Unit

Alfa 0.7 0.6{0.8 |Beta 0.5 0.4{0.6 |Lm 0.005 0.004{0.006 mg 0.002 0.0015{0.0022 m

Byp 1.8 0.6{0.8 T

with respect to the unit coe�cient). The primaryvariables: g, Lm, Beta, Alfa, Byp themselves andtheir correlation are expressed with corner elements,taking the value of +1. The correlations betweenother variables can have positive or negative valuescategorized as follows [23]:

0{0.19, no correlation;

0.2{0.39, weak correlation;

0.4{0.69, moderate correlation;

0.7{0.89, strong (high) correlation;

0.9{1, very strong correlation.

Coe�cient of determination, which is expressed asR2, identi�es how the total change in the dependentvariable can be explained by the independent variable.R2 di�ers between 0{1. Values close to 1 show thata great part of the change in dependent variables isdetermined by independent variables [23].

According to the input design variables in Table 2,the correlations between analysis process e�ciency,power out, approximate cost, and total volume arepresented in Figure 3. In the analysis performed forstroke, the maximum, peak-to-peak, and root valuesof cogging force are de�ned as output variables. Thecorrelations between input variables and stroke changeare given in Figure 2.

Figure 2. Correlation matrix between input variables and cogging force.


Figure 3. Correlation matrix between input variables and e�ciency, power out, volume, cost, and mass.

As can be seen in Figure 2, while there is amoderate negative correlation between stroke and Alfaas well as g, the correlation between stroke and Beta isweak positive.

As observed in Figure 3, there is a very strongcorrelation between mass, cost, and volume. Onthe other hand, the correlations between Beta andmass, and volume and cost are strong negative. Inaddition, there is a high positive correlation betweenBeta and e�ciency; moderate correlation between Alfaand e�ciency; strong negative correlation between gand power out; moderate correlation between Lmand power out and cost; and very weak correlationbetween Byp and mass, volume, and cost. Whilea moderate, strong, or very strong correlation withgenerator performance parameter is required by avariable to participate in optimization, there is nocorrelation between the speci�ed design variables andgenerator performance.

3. Optimization of stroke and objectivefunctions by RSM

Designers demand minimum cost when they wantmaximum e�ciency and when they want maximumperformance, they demand minimum weight. Math-ematical modeling has great bene�ts when the oppor-tunity to perform experimental work on the subject islimited or reaching results requires too many iterations.RSM, which is an application area of mathematicalmodeling, is one of the methods with partial factorialexperimental or numerical application and it is avery important statistical method for the analysis ofcorrelations between di�erent results obtained withdi�erent factors.

RSM uses statistical and experimental methodstogether. It is a group of serial processing applied to

optimizing the dependent variable among independentvariables. This method assesses independent variablesas parameters and dependent variables as responses oroutput. Tendency to maximum increase or decreasein order to reach the minimum or maximum level(maximum point desired) is the primary approach. Thee�ects of independent variables on the response can beobserved by decreasing the number of required exper-iments for optimization with RSM. While using RSM,it is important to choose the independent variableswhich are believed to in uence the dependent variablesthe most for getting correct results. The choice ofvariables can be made through a mathematical modelor correlation analysis. Primary data given in theliterature (pole pitch ratio, slot pitch, etc.) and theranges in which the variables can be measured arefound. Model equations used in RSM can be �rst-or second-degree ones. In the electric machine studiesconducted with RSM, second-degree model equationsare preferred. Eq. (1) gives RSM design variablesfunction and error:y = f (x1; x2; x3; x4; x5; � � � ; xn) + "; (1)

where y is the response of the system and xn representsthe independent variables. Here, y can be e�ciency,power out, stroke, etc. Design variables are Lm, g,alfabeta, Byp, and error ("). The di�erence betweenthe observed and expected values of y is given as theerror:y = f(x) + ": (2)

Response surface models are expressed as second-degree polynomials, which include variables and theirinteractions.

y=�0+nXi=1

�iXi+nXi=1

�iiX2i +

nXi=1

nXj=1

�ijXiXj+"0;(3)


where �0 is a �xed model; �i, �ii, and �ij are variablecoe�cients; and xi and xj are the coded independentvariables. When a second-degree response surfacemodel suitable for the variables is developed, thelevels of x1; x2; � � � ; xn, which optimize the estimatedresponse variable, should be determined. This point,which optimizes the response variable (if any), isfound by taking the partial derivatives according tothe variables x1; x2; � � � ; xn and taking them equalto zero. The design generally used to reach theoptimum point is central integrated or Box-Behnkenexperimental design. Both are second-degree modelswith second-degree terms, that is, a correlation beyondthe linear approach between independent variables andthe output values can be expressed. Figures 2 and3 represent the relationship between ANSYS Work-bench and correlation, RSM, and optimization of theparameters. In addition, design explorer establishesdesign points through input design variables de�ned inANSYS Maxwell 2D rz and given in Table 2. ANSYSMaxwell 2D rz tool is added in ANSYS Workbench. Bymeans of this tool, the primary sizes of the generator,limiting conditions, and analysis parameters are de-�ned. Then, the parameter correlation tool is added toWorkbench and correlation matrices between input andoutput variables are obtained. RSM tool is added andthe solution is realized. Here, MOGA is chosen as theoptimization method. The data obtained give the bestresult separately for each objective function (Table 3).

Table 3. Objectives of design.

Design Parameter ObjectiveType Target

Design 1 � Maximize {X1 = 1

Pout = 600 Seek target 600X2 = 0X3 = 0

Design 2 1V Maximize {

X1 = 0Pout = 600 Seek target 600X2 = 1

X3 = 0

Design 3 1C Maximize {


X3 = 0

Design 4 �V:J Maximize {


X3 = 1

The best results for each design (Table 4) are analyzedagain in ANSYS Maxwell 2D rz. The obtained FEMresults and the analytical data are compared in Table 5.Figure 4 gives ANSYS Workbench and Maxwell 2D aswell as the analysis and optimization process.

It is possible to use the MOGA option for bothresponse surface optimization and direct optimization.In this study, it was used with response surface opti-mization. With MOGA, we can generate a new sampleset or use an already existing set to provide an approachmore re�ned than the screening method. It can dealwith multiple goals and can be used for all types ofinput parameters. A fast and non-dominated sortingmethod, which is an order of magnitude faster thanthe conventional Pareto ranking methods, is Paretoranking scheme. Lagrange multipliers and penaltyfunctions are not required in this method, since con-straint handling makes use of the same non-dominanceprinciple with the same objectives. This makes itpossible to rank feasible solutions higher than infeasibleones. In a separate sample set, the �rst Pareto frontsolutions are archived distinctly from the evolving sam-ple set, ensuring that the Pareto front patterns alreadyavailable from earlier iterations get minimal disruption.By changing the maximum allowable Pareto percentageproperty, the selection pressure (and, consequently,the elitism of the process) can be controlled to avoidpremature convergence [24]. Figure 5 gives Multi-Objective Genetic Algorithm (MOGA) ow diagram.

The initial population is employed to operatethe MOGA algorithm. When MOGA is operated,it produces a new population through cross-over andmutation. Following the �rst iteration, each populationis operated after having reached the number of samplesspeci�ed by the number of samples per iteration prop-erty. MOGA proceeds to generate a new population.In the new population, design points are updatedand optimization is validated for convergence. Wheneither the maximum allowable Pareto percentage orthe convergence stability percentage has been reached,convergence of MOGA takes place. The processcontinues to the next step if the convergence of MOGAdoes not take place. Optimization is validated forful�llment of stopping criteria if it does not converge.In case of meeting the criterion of maximum numberof iterations, the process stops without having reachedconvergence. However, if the stopping criteria are notmet, in order to create a new population, MOGAis regenerated. MOGA generates a new populationthrough stopping criteria. Until the convergence ofoptimization or ful�llment of the stopping criteria,validation is repeated in sequence. The optimizationconcludes if either of these cases occurs [23]. Accordingto the graph in Figure 6, increase in Alfa and Betaincreases e�ciency and power out. An increase in Lmincreases both power out and cost. Increase in Beta will


Figure 4. Response Surface Methodology (RSM) and optimization analysis process of tubular linear generator for anobjective function with ANSYS Workbench.

Table 4. Comparison of analytical and numerical optimization results.

Input variableInitial design Design 1 Design 2 Design 3 Design 4

Analytical FEM FEM FEM FEM

Lm 0.005 0.00403 0.005524 0.0041038 0.005275

Beta 0.5 0.501 0.59486 0.59403 0.59913

Alfa 0.7 0.79033 0.63315 0.68248 0.68333

G 0.002 0.002058 0.0015203 0.0015147 0.0017788

Byp 1.8 1.815 2.1121 2.0857 2.1652

Table 5. Response Surface Methodology (RSM) optimization results.

Output parametersInitial design Design 1 Design 2 Design 3 Design 4

Analytical FEM FEM FEM FEM

Moving weight (kg) 2.38 2.51 2.37 2.55 2.39

Total weight (kg) 19.09 19.6 14.64 15.05 14.74

Approximate cost (TL) 405.6 406.1 348.9 331.25 357.82

Power out (VA) 579.7 589.9 589 582 601

E�ciency 0.856 0.88 0.82 0.85 0.841

decrease weight, cost, and volume. Finally, an increasein Alfa also increases cost.

Analyses showed signi�cant e�ects of the correla-tion between Beta and Alfa on e�ciency, weight, andcost. Thus, it is important to determine the Beta and

Alfa rates correctly for tubular and at linear machines.For initial parameters, it is suitable to choose Alfabetween 0.68 and 0.73 and Beta between 0.45 and 0.55.The change in magnet thickness with Alfa in uencese�ciency and cost signi�cantly. Magnet thickness


Figure 5. Multi-Objective Genetic Algorithm (MOGA) ow diagram.

Figure 6. Local sensitivity changes of the basic variablesin terms of output parameters.

di�ers with the magnetic equivalent circuit. Whilethe thickness of the magnet decreases demagnetizationrisk, it also causes the core material to reach magneticsaturation. During the process of optimization byRMS, while bene�t is derived from a feature, anotherfeature is withdrawn. The conducted analyses showedthat the change between magnet thickness and Betawas in the form of IF function (Figure 7). Thereis local minimum and maximum in the IF function.The optimum point of Beta can be taken as 0.5.

While the magnet thickness and change of Beta donot in uence e�ciency, signi�cantly, they a�ect totalgenerator weight and cost.

Figure 2 gives input variables and limits. Objec-tive functions given in Eqs. (4) and (5) are de�ned sepa-rately. These objective functions are given as Design 1e�ciency maximum, Design 2 volume minimum, De-sign 3 cost minimum, and Design 4 general performanceboost (E�ciency=(Volume�Current density)), as givenin Table 3. For MOGA optimization, the power out foreach objective function is 600 VA:

A.F1 =�X1

V X2 :JX3and Pout = 600; (4)

A.F2 =�X1

MX2 :JX3and Pout = 600; (5)

A.F3 =�X1

CX2 :JX3and Pout = 600; (6)

where A.F1, A.F2, and A.F3 are objective functions; �is e�ciency; V is volume; M is weight; C is cost; J iscurrent density; and Pout is output power in the designdecided by the desired values of X1, X2, and X3. Here,in order to simplify the formula, the values of X1, X2,and X3 are equalized to 0 or 1. The simpli�ed versionsof objective functions given in Eqs. (4), (5), and (6) forthe values of 0 and 1 for X1, X2, and X3 and the targetare given in Table 3. Here, two objectives are de�ned


Figure 7. The in uence of Lm-alfa-beta change one�ciency.

for each design in MOGA to maximize power out andfunctions for all targets.

Since the amount of the desired nominal valuedoes not in uence the optimization result in MOGAoptimization method, the best proposed sizing dataamong the proposed 4 nominal values are given inTable 4.

Table 5 shows how much increase or decreasethe obtained data show in terms of weight, cost, ande�ciency in comparison with the initial design data.Here, while Design 1 shows 2.6% increase in the total

Figure 8. The distribution of the magnetic ux densityon the length of the generator in ANSYS Maxwell 2D rz.

mass and no signi�cant di�erence in cost, it provides2.8% increase in e�ciency. In addition to 23.3%decrease in the total mass and 13.9% decrease in cost,Design 2 shows 3.5% decrease in e�ciency. With 21.1%decrease in the total mass and 18.3% decrease in cost,Design 3 makes no signi�cant di�erence in e�ciency.Finally, Design 4 leads to 22.7% decrease in the totalmass and 11.78% decrease in cost. According to theanalytical and numerical calculation results given inTable 5 for RMS optimization, the highest e�ciency isseen in Design 1.

Results are given in Table 4, showing a goodagreement between the FEM calculations and theanalytically calculated values for the optimum design.The most suitable situation is seen with Design 4, inwhich there is a signi�cant decrease in mass and costand very little change in e�ciency. In line with thedesign data given in Table 1, calculations of lineargenerator were made in the interface [19]. However,when the size of the magnet obtained from MOGAresults was considered, it was di�cult to producemagnets, speci�cally. Thus, generator geometry wasestablished based on the initial design geometry data.

The distribution of the magnetic ux density onthe length of the generator in ANSYS Maxwell 2Drz can be seen in Figure 8. The images of crankslider mechanism are given in Figure 9(a), (b), and (c).M43-24G geometry and magnetic rotor piece with ironsheet for the prototype are given in Figure 9(d) and(e). The prototype machine can be seen in Figure 10,which is fabricated based on the initial design. InFigure 10, the generator is driven by the crank slidermechanism (Figures 9(b) and 10(b)) proper to a 4-poleasynchronous motor.

The results of the numerical analysis were com-pared with the testing prototype (unloaded) for 20 Hzdriving frequency (Figure 11). Here, speed was calcu-lated according to the crank sizes given. It was foundthat the results of the numerical analysis by ANSYSMaxwell and those of the application were in parallelto a great extent. In the experimental study, nominalworking speed and frequency values were not reacheddue to mechanic vibration.


Figure 9. (a) Slider-crank mechanism dimensions: r: radius of crank, l: length of connecting rod, and �: rank angle. (b)Crank slider mechanism view in application. (c) Measurement of length of connecting rod. (d) Parts of the generator. (e)Generator view of the prototype.

Figure 10. (a) Cranck slider mechanism. (b) Tubular linear permanent-magnet generator test rig.

Figure 11. (a) Speed crank angle change. (b) The voltage induced in open-circuit phase winding.


4. Conclusion

Nowadays, studies on free-piston motor-generator sys-tems as range increasing units for electrical or hybridvehicles have become very important. In free-pistonmotor-generator systems, it is vital to increase thee�ciency of the electric energy produced in returnfor the fuel consumed. Thus, new approaches andmethods for making the generator/motor design inthese systems compact, light, and highly e�cient arevery important. In addition, since decreasing movingweight will increase mechanical frequency, it has adirect in uence on generator performance.

In this study, sizing optimization was performedfor free piston practices through Response SurfaceMethodology (RSM) by the analytical results of thetubular linear generator model. We tried to reachthe constraint values by making continuous iterationsin line with sizing equations and objective functions.These functions were aimed at increasing e�ciency anddecreasing cost, volume, and total mass. Objectivefunctions determined through RSM were applied to theMulti-Objective Genetic Algorithm (MOGA) analysis.In addition, it was found that Design 4, which includedgeneral performance, did not cause signi�cant changein moving weight and led to decreases by 1.75% ine�ciency, 22.7% in mass, and 11.78% in cost. It wasshown that RSM and the optimization method, whichcomprised MOGA, were successfully implemented.

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Appendix

The application of the cosine theorem according to thetriangle in the moving system in Figure 9(a) is:

l2 = x2 + r2 � 2xr cos�; (A.1)

x2 � 2xr cos� = l2 � r2; (A.2)

x2�2xr cos�+r2 cos2 �= l2�r2+r2 cos2 �; (A.3)

(x� r cos�)2 = l2 � r2(1� cos2 �); (A.4)

where:

sin2 � = 1� cos2 �: (A.5)

By taking the square root of two sides of the equality:

x� r cos� =pl2 � r2 sin2 �; (A.6)

x = r cos�+pl2 � r2 sin2 �: (A.7)

l is length and r a crank radius, which is constant.

The crank angle (�) varies between (0�{360�) and x isthe only variable that a�ects the piston position. At� = 0�, the piston is in the top point and position sizeis l+ r. At � = 180�, the piston is in the bottom pointand position size is l � r. Piston speed is a derivativeof displacement:

v =dxdt: (A.8)

Angular velocity is:

! =d�dt; (A.9)

v =dxd�� d�dt; (A.10)

v =dxd�!: (A.11)

If the position of Eq. (A.7) is derived according to thealpha:

dxd�

=� r sin�+12�l2 � r2 sin2 �

��1=2 dd��

l2 � r2 sin2 ��; (A.12)

dxd�

=� r sin�+1

2pl2 � r2 sin2 ��

0� r22 sin� cos��; (A.13)

dxd�

= �r sin�+r2 sin� cos�pl2 � r2 sin2 �

: (A.14)

According to Eqs. (A.11) and (A.14), the speed can becalculated as follows:

v =

"�r sin�� r2 sin� cos�p

l2 � r2 sin2 �

#x!: (A.15)

Biographies

Serdal Arslan was born in Bolu, Turkey, in 1983.He received the MSc and PhD degrees in ElectricalEducation from the Institute of Natural and AppliedSciences, University of Gazi, Ankara, Turkey, in 2011and 2017, respectively. He worked as a ResearchAssistant at Hakkari University between 2010 and2011. Then, he attended Harran University, where heis currently a lecturer in the Department of ElectricalTechnology. He has been working for many yearsin the �eld of computational electromagnetics. Hisresearch interests are in the areas of design, mod-elling, and analysis of electrical machines; optimizeddesign of linear/rotary machines; and electromagneticdevices.


Osman G�urdal received the BSc and MSc degreesin Electrical Education from the Technical EducationFaculty of the University of Gazi, Ankara, Turkey, in1985 and 1991, respectively. He received the PhDdegree in Electrical and Computer Engineering fromthe University of Sussex, Brighton, UK, in 1994.After 27 years of academic life at the University ofGazi, he is now professor at the University of BursaOrhangazi. His research interests are computer aideddesign, analysis, and simulation of electrical machines,

magnetic circuits, and actuators as well as �eld the-ory.

Sibel Akkaya Oy received the BSc degree in Electri-cal Education from Dicle University, Turkey, in 2005and the MSc degree in the same �eld from FiratUniversity, Turkey, in 2007. Also, she received the PhDdegree in Electrical Education from Gazi University,Turkey, in 2014. Her research focuses on the electricalmachine design and renewable energy.

design and optimization of tubular linear permanent-magnet

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